N-representability of diagonal elements of second-order reduced density matrices

The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x₁, x₂) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x₁, x₂) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θ_n(x₁, x₂) converging at n → ∞ to ρ(x₁, x₂)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x₁, x₂). The results obtained may open new possibilities for using ρ(x₁, x₂) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 127–142, 1997     

Ab initio calculations of many-body energy expansion in LiF− clusters

The effects of the basis-set size on many-body energy expansion in LiF^? clusters are investigated and correlated with previously reported values on LiCl^? analogs. Coulomb and non-Coulomb energies in LiF^? at different configurations are also examined. Although at the minimal STO -3G basis V^na(3, 4) and V^na(4, 4) nonadditivity terms were the smallest in the D₃h configuration, they were the largest at the extended 6-311 ++G basis. V(m, n) terms where m = n ≥ 3 were found to be playing a small role in the chemistry and physics of LiF^? clusters compared with V(3, n) terms in LiCl^? clusters.     

Control of Oxidation States of Transition Elements (Cr,Mn) in Nitridometalates and the Solid Solution Series Li6(Ca1‐xSrx)2[Mn2N6]

The formation of ternary nitridometalates from the elements in the case of the systems Li—Cr, V, Mn—N leads to compounds which contain the transition metals in the highest (V^V, Cr^VI) or a comparably high (Mn^V) oxidation state. In the corresponding calcium and strontium systems, the transition metals show a lower oxidation state (V^III, Cr^III, Mn^III). Transition metals with intermediate oxidation states (Cr^V, Mn^IV) are present in the quaternary (mixed cation) compounds Li₄Sr₂[CrN₆], Li₆Ca₂[MnN₆], and Li₆Sr₂[MnN₆] (R3¯(#148), a = 585.9(3) pm, c = 1908.6(4) pm, Z = 3), as well as in the solid solution series Li₆(Ca_1—xSr_x)₂[MnN₆].     

Slater's nonlocal exchange potential and beyond

The local density approximation (LDA) to the exchange potential V_x( r ), namely the ρ^1/3 electron gas form, was already transcended in Slater's 1951 paper. Here, using Dirac's 1930 form for the exchange energy density ? _x( r ), the Slater (Sl) nonlocal exchange potential V( r ) is defined by 2? _x( r )/ρ( r ). In spherical atomic ions, say the Be or Ne‐like series, this form V( r ) already has the correct behavior in both r → 0 and r → ∞ limits when known properties of the exchange energy density ? _x( r ) and the ground‐state electron density ρ( r ) are invoked. As examples, some emphasis will first be given to the use of the so‐called 1/Z expansion in such spherical atomic ions, for which analytic results can be obtained for both ? _x( r ) and ρ( r ) as the atomic number Z becomes large. The usefulness of the 1/Z expansion is directly demonstrated for the U atomic ion with 18 electrons by comparison with the optimized effective potential prediction. A rather general integral equation for the exchange potential is then proposed. Finally, without appeal to large Z, two‐level systems are considered, with specific reference to the Be atom and to the LiH molecule. In all cases treated, the Slater potential V( r ) is a valuable starting point, even though it needs appreciable quantitative corrections reflecting directly atomic shell structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Ladder operators for the Kratzer oscillator and the Morse potential

Taking a close look at the Infeld–Hull ladder operators for the Kratzer oscillator system, V(x) = [x² + β(β ? 1)x^?2]/2, we deduce and explicitly construct energy‐raising and ‐lowering operators for the generalized Morse potential system V(z) = (Ae^?4αz ? Be^?2αz)/2, through a canonical transformation that exists between the two systems. For the Morse potential system, we obtain a system of raising and lowering operators P_±(n) (n = 0, 1, 2, 3, … , n_max) with the specific property that P_±(n)Φ_n = c_±(n)Φ_n±1, where Φ_n denotes the nth energy eigenfunction. While P_?(0) annihilates the ground‐state Φ₀, the operator P₊(n_max), instead of annihilating the highest bound‐state Φ, actually knocks it out of the L₂ space spanned by the discrete bound states and becomes inadmissible. Yet, raising and lowering operators _± with proper end‐of‐spectrum behavior (i.e., _?|0〉 = 0 and ₊|n_max〉 = 0) can be constructed in a straightforward way in the energy representation. We show that the operators ₊, _?, and ₀ (where ₀ ≡ (1/2)[ ₊, _?]) form a su(2) algebra only if we restrict them to the (N ? 1)‐dimensional subspace spanned by the lowest (N ? 1) basis vectors, but not in the full (N + 1)‐dimensional space spanned by the discrete bound states [N ≡ n_max ≡ integral part of (1/2)(B/(2α ) ? 1)]. Realization of this su(2) algebra in the position representation (when restricted to the (N ? 1)‐dimensional subspace) is also given. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Generalized morse analytic function for the “true” diatomic potential of the RKR type

The problem of the representation of the RKR (or IPA) diatomic potential by a simple analytic function is considered. This old problem has for a fairly good solution the Coxon-Hajigeorgiou function U(x) = D[1 - exp-f_n(x)]² with f_n(x) = Σ a_mx^m. The problem of the determination of the disposable parameters a₁ … a_n [in order that U(r) fits the given RKR potential] is reduced to that of a set of linear equations in a_m where a standard least-squares technique is used. The application to several states (ground or excited) of several molecules shows that a fairly “good” fit is obtained for n ～ 10, even for the state XO_g—I₂ bounded by 109 vibrational levels, for which the RKR potential is defined by the coordinates of 219 points. It is shown that the percentage deviation |U(r)^RKR - U(r)| throughout the range of r values is about 0.04% for XΣ? Li₂, 0.0005% for XΣ? HCl, 0.06% for XO_g? I₂, and 0.05% for BO_u? I₂ (as examples). This approach shows the same success for deep and shallow potentials. The comparison of the computed E_v (vibrational energy) and B_v (rotational constant) with their corresponding experimental values shows that a good agreement is reached even for high vibrational levels close to the dissociation. © John Wiley & Sons, Inc.     

The N(4S) + N2O(X̃1∑) reaction

The title reaction, which is spin‐forbidden for N₂(X¹∑) + NO(X²Π) production, has been studied from 960 to 1130 K in a high‐temperature photochemistry reactor. No reaction could be observed, indicating k < 1 × 10^?15 cm³ molecule^?1 s^?1. It is concluded that there is no significant contribution from the spin‐allowed exothermic path leading to N₂(X¹∑) + NO(a⁴Π). © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 387–389, 2001     

Studies on the composition and kinetics of chromium(III)-alanine system

Kinetics of the complex formation of chromium(III) with alanine in aqueous medium has been studied at 45, 50, and 55°C, pH 3.3–4.4, and μ = 1 M (KNO₃). Under pseudo first-order conditions the observed rate constant (k_obs) was found to follow the rate equation: Values of the rate parameters (k_an, k, K_IP, and K) were calculated. Activation parameters for anation rate constants, ΔH^≠(k_an) = 25 ± 1 kJ mol^?1, ΔH^≠(k) = 91 ± 3 kJ mol^?1, and ΔS^≠(k_an) = ?244 ± 3 JK^?1 mol^?1, ΔS^≠(k) = ?30 ± 10 JK^?1 mol^?1 are indicative of an (I_a) mechanism for k_an and (I_d) mechanism for k routes (‥substrate Cr(H₂O) is involved in the k route whereas Cr(H₂O)₅OH²⁺ is involved in k′ route). Thermodynamic parameters for ion-pair formation constants are found to be ΔH°(K_IP) = 12 ± 1 kJ mol^?1, ΔH°(K) = ?13 ± 3 kJ mol^?1 and ΔS°(K_IP) = 47 ± 2 JK^?1 mol^?1, and ΔS°(K) = 20 ± 9 JK^?1 mol^?1.     

The density matrix in the two-particle function method

The first- and second-order density matrices D ^(N) and D for the function g⁽ⁿ⁾ = A_N[g(1, 2) …? g(N ? 1, N)] are expressed by the g function itself and its density matrix D . In a singlet state the generating functions for spatial parts of these matrices are simply connected with there solvent of the Fredholm equation in which the spatial part of D is a kernel. Some special cases of g(1, 2) are considered. It isestablished that the number of large eigenvalues of D does not exceed that of different eigenvalues of D . Thus the degeneracy in the spectrum of D causes the appearance of such large eigenvalues.     

Density functional theory of the iron–nitrosyl (S = 3/2) complex

On the basis of density functional theory (DFT), the iron–nitrosyl complex Fe[Me₃TACN](NO)(N₃)₂ (S = 3/2) is studied via the B3LYP hybrid method. Its Raman vibrational frequencies, atomic net charges, and spin densities are analyzed. The related complexes Fe(NH₃) (n = 1, 2, and 3) are employed as reference compounds to determine the characteristics of the central iron. Our results indicate that the S = 3/2 spin ground state of Fe[Me₃TACN](NO)(N₃)₂ is best described by the presence of Fe^II (S = 2) anti‐ferromagnetically coupled to NO⁰ (S = 1/2) yielding Fe^II[Me₃TACN](NO⁰)(N)₂. This is clearly different from the previous Fe^III‐NO^? theoretical assignment. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Rate constants and conversion ratios for aqueous‐phase reactions of SO with CnF2n+1C(O)O− (n = 4–7) at 298 K

Kinetic data for aqueous‐phase reactions of sulfate anion radicals (SO) with perfluorocarboxylates (C_nF_2n+1C(O)O^?) are needed to evaluate removal and transformation processes of C_nF_2n+1C(O)O^? species in the environment, but rate constants for the reactions of SO with C_nF_2n+1C(O)O^? (k_n) have been reported only for short‐chain C_nF_2n+1C(O)O^? species (n = 1–3). Since C_nF_2n+1C(O)O^? reacted with SO to form C_mF_2m+1C(O)O^? (m < n), we determined relative rates k_n?1/k_n for the reactions of SO with C_nF_2n+1C(O)O^? (n = 4–7), along with conversion ratios for conversion of C_nF_2n+1C(O)O^? into C_n?1F_2n?1C(O)O^? (α_n) and into C_n?2F_2n?3C(O)O^? (β_n) at 298 K. SO was photolytically generated from S₂O by use of sunlamps (λ ≈ 310 nm). Even if k_n and k_n?1 change with increasing ionic strength, k_n?1/k_n can be determined when k_n?1/k_n and α_n remain almost constant during the reaction. The values of k_n/k₁ for n = 4–7 were nearly equal, and their average was 0.82 ± 0.04 (2σ). Conversion ratios of α_n and β_n were mostly independent of n for n = 4–7, and their averages were 0.77 ± 0.07 (2σ) and 0.13 ± 0.08, respectively. Branching ratios of reactions of a possible intermediate (C_nF_2n+1O^?), reaction of C_nF_2n+1O^? with H₂O, and fission of the C? C bond of C_nF_2n+1O^?, seemed to determine α_n and β_n. © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 735–747, 2009     

The heats of formation of NO3− and NO3− association complexes with HNO3 and HBr

The equilibrium constant for the reaction has been determined between 331 and 480°K using a variable-temperature flowing afterglow. These data give ΔH°(1) = -1.03 ± 0.21 kcal/mol and ΔS°(1) = —4.6 ± 1.0 cal/mol°K. When combined with the known thermochemical values for HBr, Br^?, and HNO₃, this yields ΔH(NO₃^?) = -74.81 ± 0.54 kcal/mol and S(NO₃^?) = 59.4 cal/mol·°K. In addition ΔH_n-1,n and ΔS_n-1,nfor the gas-phase reactions were determined for n = 2 and 3. The implications of these measurements to gas-phase negative ion chemistry are discussed.     

The reactions of polyacetylene with trimethyloxonium hexachloroantimonate. Covalent doping of (CH)x

Polyacetylene, (CH)_x, has been doped with trimethyloxonium hexachloroantimonate, (CH₃)₃O⁺SbCl(1), in dichloromethane and acetonitrile. The maximally doped (CH)_x films have moderate conductivities [σ_RT(CH₂Cl₂) = 10, σ_RT(CH₃CN) = 0.7 Ω^?1 cm^?1]. Reactions between 1 and (CH)_x CH₂Cl₂ or CH₃CN were followed in situ by ¹H nuclear magnetic resonance spectroscopy and x-band electron spin resonance spectroscopy. It was found that the reactions in the two solvents are different. In dichloromethane the dopant is SbCl₅, which forms from the decomposition of 1, and doping proceeds by electron removal from (CH)_x chains. Based on the ESR signal loss, an estimate can be made of the diffusion rate of SbCl₅, into the (CH)_x fibrils in CH₂Cl₂; it is found to be ca. 10^?17 cm²/s. In acetonitrile the dopant appears to be either CH₃CNCH, H⁺, CH, or a combination of one or more of these dopants. It is postulated that the CH₃CNCH, CH, and/or H⁺ dopant covalently binds to the (CH)_x chain. X-ray photoelectron spectra show that films doped with excess 1 in both solvents have approximately one SbCl per 33 CH units.     

Applications of n-electron wave functions built up with two kinds of geminals

The ground states of atoms and molecules Li^?, Be, LiH, LiH, and Li₂ have been calculated using the n-electron wave functions built up with two kinds of geminals. As a comparison, the above systems have been calculated with the Hartree–Fock self-consistent field and the multi-configuration self-consistent field method as well. The results show that the wave functions in this work are capable of describing the electron correlations, and both kinds of geminals can be taken as a starting point in building up n-electron ground states.     

Exploring sequential quantum adiabatic switching across supersymmetric partners for finding the eigenstates of a system

We demonstrate that one can exhaustively determine the n‐bound eigenstates of a Hamiltonian H by constructing a sequence of supersymmetric (SUSY) partner Hamiltonians and invoking a time‐dependent quantum adiabatic switching algorithm for passage from the ground state of one to the other. The ground states of the initial pair H⁽⁰⁾ and H⁽¹⁾ are constructed by solving the Riccati equation for the superpotential ?⁽⁰⁾ for H⁽⁰⁾ and adiabatically switching from the ground state Ψ of H⁽⁰⁾ to the ground state Ψ of H⁽¹⁾. The charge operator Q is then used to recover the first excited state Ψ of H⁽⁰⁾. The procedure is repeated for the ground states of SUSY pairs H^{(n + 1)} and H^{(n + 2)}, and appropriate charge operators lead to the excited states Ψ of H⁽⁰⁾ with , thereby exhausting the full eigenspectrum of H⁽⁰⁾. The workability of the proposed method is shown with several well‐known examples. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Ab initio characterization of size dependence of electronic spectra for linear anionic carbon clusters C(n) (-) (n = 4-17)

In this article, we determine the ground‐state equilibrium geometries of the linear anionic carbon clusters C (n = 4–17) by means of the density functional theory B3LYP, CAM‐B3LYP, and coupled cluster CCSD(T) calculations, as well as their electronic spectra obtained by the multireference second‐order perturbation theory CASPT2 method. These studies indicate that these linear anions possess doublet ²∏_g or ²∏_u ground state, and the even‐numbered clusters are generally acetylenic, whereas the odd‐numbered ones are essentially cumulenic. The energy differences, electron affinities, and incremental binding energies of C chains all exhibit a notable tread of parity alternation, with n‐even chains being more stable than n‐odd ones. In addition, the predicted vertical excitation energies from the ground state to four low‐lying excited states are in reasonably good agreement with the available experimental observations, and the calculations for the higher excited electronic transitions can provide accurate information for the experimentalists and spectroscopists. Interestingly, the absorption wavelengths of the 1²∏_u/g ← X²∏_g/u transitions of the n‐even clusters show a nonlinear trend of exponential growth, whereas those of the n‐odd counterparts are found to obey a linear relationship as a function of the chain size, as shown experimentally. Moreover, the absorption wavelengths of the transitions to the higher excited states of C series have the similar linear size dependence as well. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

A finite-difference solution of the Hartree–Fock equations for diatomic molecules

A more general application of the self-consistent field iteration is coupled with a finite-difference Newton–Raphson algorithm to solve the set of coupled second-order integro-differential equations with split boundary conditions which constitutes the Hartree–Fock problem for diatomic molecules. The N orbitals are assumed to be of the form ψ_α = L_α(λ) M_α (μ)e^imα^? (2π)^?1/2, (α = 1, …?, N), where λ, μ, and ? are the usual confocal elliptical coordinates. Requiring the expectation value of the electronic Hamiltonian to be stationary with respect to independent variations of the functions L_α and M_α, subject to constraints of orthonormality, leads to a set of coupled one-dimensional differential equations for the functions L_α and M_α. In the new method a corresponding set of finite-difference equations including the split boundary conditions for each function, as well as the Lagrange multipliers and associated constraints on normalization and orthogonality, are incorporated into a large system of nonlinear algebraic equations which is solved by means of a coupled self-consistent field-generalized Newton–Raphson iteration. As examples, calculations of the (1)^{2 1}Σ and (1) (2pσ_u) ³Σ states of H₂ are presented. The calculated energy for the ¹Σ state of H₂ is 99.985% of the three-dimensional Hartree–Fock limit. The discrepancy is due to the assumed factored form of the orbitals ψ_α, and a generalization of the finite-difference method is suggested to improve the results.     

Melt‐state polymer chain dimensions as a function of temperature

The unperturbed chain dimensions (〈R²〉_o/M) of cis/trans‐1,4‐polyisoprene, a near‐atactic poly(methyl methacrylate), and atactic polyolefins were measured as a function of temperature in the melt state via small‐angle neutron scattering (SANS). The polyolefinic materials were derived from polydienes or polystyrene via hydrogenation or deuteration and represent structures not encountered commercially. The parent polymers were prepared via lithium‐based anionic polymerizations in cyclohexane with, in some cases, a polymer microstructure modifier present. The polyolefins retained the near‐monodisperse molecular weight distributions exhibited by the precursor materials. The melt SANS‐based chain dimension data allowed the evaluation of the temperature coefficients [dln 〈R²〉_o/dT(κ)] for these polymers. The evaluated polymers obeyed the packing length (p)‐based expressions of the plateau modulus, G = kT/np³ (MPa), and the entanglement molecular weight, M_e = ρN_anp³ (g mol^?1), where n_t denotes the number (～21) of entanglement strands in a cube with the dimensions of the reptation tube diameter (d_t) and ρ is the chain density. The product np³ is the displaced volume (V_e) of an entanglement that is also expressible as pd or kT/G. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1768–1776, 2002     

Potential energy surfaces of carbon dioxide

Several bent valence states of CO₂ are characterized by means of full-valence-space MCSCF calculations. The ground state potential energy surface exhibits a double well corresponding to a ring minimum, with C_2vsymmetry (¹A₁) and a 73.1° OCO angle, in addition to the linear (¹σ) global minimum. The transition state for the ring opening process, which has a barrier of 12.1 kcal/mole with respect to the ring minimum, is however found to have C_s symmetry. Double minima are also shown to exist for the ¹A₂, ¹B₁ and ¹B₂ excited states. However, in these cases all minima are bent. Cross sections through the ground state potential energy surface corresponding to the two collinear exchange reactions O(¹D) + CO(¹σ⁺) → OC(¹σ⁺) + O(¹D) C(³P) + O₂(³σ) → CO(¹σ⁺) + O(¹D) are also calculated and their energy contour maps are reported. The latter reveals the existence of a stable linear intermediate with the structure COO. © 1994 John Wiley & Sons, Inc.     

Sputtered ammonium dinitramide: Tandem mass spectrometry of a new ionic nitramine

The recently synthesized ammonium dinitramide (ADN) is an ionic compound containing the ammonium ion and a new oxide of nitrogen, the dinitramide anion (O₂N? N? NO₂^?). ADN has been investigated using high-energy xenon atoms to sputter ions directly from the surface of the neat crystalline solid. Tandem mass spectrometric techniques were used to study dissociation pathways and products of the sputtered ions. Among the sputtered ionic products were NH₄⁺, NO⁺, NO₂^?, N₂O₂^?, N₂O, N₃O₄^? and an unexpected high abundance of NO₃^?. Tandem mass spectra of the dinitramide anion reveal the uncommon situation where a product ion (NO₃^?) is formed in high relative abundance from metastable parent ions but is formed in very low relative abundance from collisionally activated parent ions. It is proposed that the nitrate anion is formed in the gas phase by a rate-determining isomerization of the dinitramide anion that proceeds through a four-centered transition state. The formation of the strong gas-phase acid, dinitraminic acid (HN₃O₄), the conjugate acid of the dinitramide anion, was observed to occur by dissociation of protonated ADN and by dissociation of ADN aggregate ions with the general formula [NH₄(N(NO₂)₂)_n] NH₄⁺, where n = 1–30.